Squared Weil and Tate Pairing Techniques for Use with Elliptic Curves

ABSTRACT

Methods and apparati are provided for use in determining “Squared Weil pairings” and/or “Squared Tate Pairing” based on an elliptic curve, for example, and which are then used to support cryptographic processing of selected information. Significant improvements are provided in computing efficiency over the conventional implementation of the Weil and Tate pairings. The resulting Squared Weil and/or Tate pairings can be substituted for conventional Weil or Tate pairings in a variety of applications.

RELATED PATENT APPLICATIONS

This patent application claims priority to commonly assigned co-pendingpatent application Ser. No. 10/626,948 (Attorney's Docket No.MS1-1276US) entitled “Squared Weil and Tae Pairing Techniques for usewith Elliptic Curves” to Eisentraeger et al., to be issued as U.S. Pat.No. 7,298,839 on Nov. 20, 2007, which are incorporated by referenceherein for all that it teaches and discloses.

This patent application is related to commonly assigned co-pendingpatent application Ser. No. 10/627,281 (Attorney's Docket No.MS1-1275US), entitled “Improved Weil and Tate Pairing Techniques UsingParabolas” to Eisentraeger et al., filed on Jul. 25, 2003, which areincorporated by reference herein for all that it teaches and discloses.

TECHNICAL FIELD

This invention relates to cryptography, and more particularly to methodsand apparati that implement improved processing techniques for Weil andTate pairings using elliptic curves.

BACKGROUND

As computers have become increasingly commonplace in homes andbusinesses throughout the world, and such computers have becomeincreasingly interconnected via networks (such as the Internet),security and authentication concerns have become increasingly important.One manner in which these concerns have been addressed is the use of acryptographic technique involving a key-based cipher. Using a key-basedcipher, sequences of intelligible data (typically referred to asplaintext) that collectively form a message are mathematicallytransformed, through an enciphering process, into seeminglyunintelligible data (typically referred to as ciphertext). Theenciphering can be reversed, allowing recipients of the ciphertext withthe appropriate key to transform the ciphertext back to plaintext, whilemaking it very difficult, if not nearly impossible, for those withoutthe appropriate key to recover the plaintext.

Public-key cryptographic techniques are one type of key-based cipher. Inpublic-key cryptography, each communicating party has a public/privatekey pair. The public key of each pair is made publicly available (or atleast available to others who are intended to send encryptedcommunications), but the private key is kept secret. In order tocommunicate a plaintext message using encryption to a receiving party,an originating party encrypts the plaintext message into a ciphertextmessage using the public key of the receiving party and communicates theciphertext message to the receiving party. Upon receipt of theciphertext message, the receiving party decrypts the message using itssecret private key, and thereby recovers the original plaintext message.

The RSA (Rivest-Shamir-Adleman) method is one well-known example ofpublic/private key cryptology. To implement RSA, one generates two largeprime numbers p and q and multiplies them together to get a largecomposite number N, which is made public. If the primes are properlychosen and large enough, it will be practically impossible (i.e.,computationally infeasible) for someone who does not know p and q todetermine them from knowing only N. However, in order to be secure, thesize of N typically needs to be more than 1,000 bits. In somesituations, such a large size makes the numbers too long to bepractically useful.

One such situation is found in authentication, which can be requiredanywhere a party or a machine must prove that it is authorized to accessor use a product or service. An example of such a situation is in aproduct ID system for a software program(s), where a user musthand-enter a product ID sequence stamped on the outside of the properlylicensed software package as proof that the software has been properlypaid for. If the product ID sequence is too long, then it will becumbersome and user unfriendly.

Additionally, not only do software manufacturers lose revenue fromunauthorized copies of their products, but software manufacturers alsofrequently provide customer support, of one form or another, for theirproducts. In an effort to limit such support to their licensees,customer support staffs often require a user to first provide theproduct ID associated with his or her copy of the product for whichsupport is sought as a condition for receiving support. Many currentmethods of generating product IDs, however, have been easily discernedby unauthorized users, allowing product IDs to be generated byunauthorized users.

Given the apparent ease with which unauthorized users can obtain validindicia, software manufacturers are experiencing considerable difficultyin discriminating between licensees and such unauthorized users in orderto provide support to the former while denying it to the latter. As aresult, manufacturers often unwittingly provide support to unauthorizedusers, thus incurring additional and unnecessary support costs. If thenumber of unauthorized users of a software product is sufficientlylarge, then these excess costs associated with that product can be quitesignificant.

New curve-based cryptography techniques have recently been employed toallow software manufacturers to appreciably reduce the incidence ofunauthorized copying of software products. For example, product IDs havebeen generated using elliptic curve cryptographic techniques. Theresulting product IDs provide improved security. Curve-basedcryptographic techniques may also be used to perform other types ofcryptographic services.

As curve-based cryptosystems grow in popularity, it would be useful tohave new and improved techniques for performing the computationsassociated with the requisite mathematical operations. Hence, there is acontinuing need for improved mathematical and/or computational methodsand apparati in curve-based cryptosystems.

SUMMARY

Methods and apparati are provided for determining “Squared Weilpairings” and/or “Squared Tate pairings” based on an elliptic curve, forexample, and which are then used to support cryptographic processing ofselected information. Significant improvements are provided in computingefficiency over conventional implementations of the Weil and Tatepairings. The resulting Squared Weil and/or Tate pairings can besubstituted for conventional Weil or Tate pairings in a variety ofapplications.

The above stated needs and/or others are met, for example, by a methodthat includes selecting an elliptic curve, determining a Squared Weilpairing based on the elliptic curve, and cryptographically processingselected information based on the Squared Weil pairing.

Here, for example, an elliptic curve E over a field K may be used,wherein E can be represented as an equation y²=x³+ax+h. The method maythen include establishing a point id that is defined as a point atinfinity on E, and wherein P, Q, R, X are points on E wherein X is anindeterminate denoting an independent variable of a function, andwherein x(X), y(X) are functions mapping the point X on E to its affinex and y coordinates, and wherein a line passes through any three pointsP, Q, R which satisfy P+Q+R=id. Note, as used herein, a bold + or −operator denotes arithmetic in the elliptic curve group, whereas anormal + or − operator denotes arithmetic in the field K or in theintegers.

In certain implementations, P and Q are m-torsion points on E and m isan odd prime, and the method further includes determining the squaredWeil pairing based on

${\frac{{f_{m,P}(Q)}{f_{m,Q}( {- P} )}}{{f_{m,P}( {- Q} )}{f_{m,Q}(P)}} = {- {e_{m}( {P,Q} )}^{2}}},$

where e_(m) denotes the Weil-pairing.

Another exemplary method includes determining a Squared Weil Pairinge_(m)(P, Q)² by establishing an odd prime m and a curve E; and based ontwo m-torsion points P and Q on E, computing e_(m)(P, Q)². In certainfurther implementations, the method also includes forming a mathematicalchain for m. For example, an addition chain or an addition-subtractionchain may be formed such that every element in the mathematical chain isa sum or difference of two earlier elements in the mathematical chain,which continues until m is included.

Then, for example, for each j in the mathematical chain, a tuplet_(j)=[jP, jQ, n_(j), d_(j)] is formed such that

$\frac{n_{j}}{d_{j}} = {\frac{{f_{j,P}(Q)}{f_{j,Q}( {- P} )}}{{f_{j,P}( {- Q} )}{f_{j,Q}(P)}}.}$

In accordance with still other exemplary implementations, another methodincludes determining a Squared Weil pairing (m, P, Q), where m is an oddprime number, by setting t₁=[P, Q, 1, 1], using an addition-subtractionchain to determine t_(m)=[mP, mQ, n_(m), d_(m)], and if n_(m) and d_(m)are nonzero, then determining:

${{i.\mspace{14mu} \frac{n_{m}}{d_{m}}} = \frac{{f_{m,P}(Q)}{f_{m,Q}( {- P} )}}{{f_{m,P}( {- Q} )}{f_{m,Q}(P)}}};$

and cryptographically processing selected information based on theSquared Weil pairing.

Another exemplary method includes selecting an elliptic curve,determining Squared Tate pairing based on the elliptic curve, andcryptographically processing selected information based on the SquaredTate pairing. Here, for example, an elliptic curve E over a field K withq=p^(n) elements may be used, wherein E can be represented as anequation y²=x³+ax+b. Letting m be an odd prime dividing q−1 and P be anm-torsion point on E. Let Q≠id be a point on E. If P is not equal to amultiple of Q, the method may further include determining

${( \frac{f_{m,P}(Q)}{f_{m,P}( {- Q} )} )^{\frac{q - 1}{m}} = {v_{m}( {P,Q} )}},$

where v_(m) denotes the squared Tate-pairing.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is illustrated by way of example and notlimitation in the figures of the accompanying drawings. The same numbersare used throughout the figures to reference like components and/orfeatures.

FIG. 1 is a block diagram illustrating an exemplary cryptosystem inaccordance with certain implementations of the present invention.

FIG. 2 illustrates an exemplary system using a product identifier tovalidate software in accordance with certain implementations of thepresent invention.

FIG. 3 illustrates an exemplary process for use in curve-basedcryptosystems in accordance with certain implementations of the presentinvention.

FIG. 4 illustrates a more general exemplary computer environment whichcan be used in various implementations of the invention.

DETAILED DESCRIPTION Introduction

The discussions herein assume a basic understanding of cryptography bythe reader. For a basic introduction to cryptography, the reader isdirected to a book written by Bruce Schneier and entitled “AppliedCryptography: Protocols, Algorithms, and Source Code in C,” published byJohn Wiley & Sons with copyright 1994 (or second edition with copyright1996).

Described herein are techniques that can be used with a curve-basedcryptosystem, and in particular elliptic curve-based cryptosystems. Incertain examples, the techniques take the form of methods and apparatithat can be implemented in logic within one or more devices. One suchdevice, for example, is a computing device that is configured to performat least a portion of the processing required for a particularcryptographic capability or application.

The techniques provided herein can be implemented and/or otherwiseadapted for use in a variety of cryptographic capabilities andapplications. By way of example, the techniques may be employed tosupport: key generation logic, e.g., for one-round three-way keyestablishment applications; identity-based encryption logic; shortsignature logic, e.g., product identifier logic; and/or other likecryptographic logic.

The term logic as used herein is meant to include any suitable form oflogic that may be employed. Thus, for example, logic may includehardware, firmware, software, or any combination thereof.

The term curve-based cryptosystem as used herein refers to logic that atleast partially provides for curve-based encryption and/or decryptionusing key(s) that are generated based at least partially on aspects orcharacteristics of an elliptic curve or other like curve.

Such curve-based cryptosystems can be used to encrypt any of a widevariety of information. Here, for example, one exemplary cryptosystem isdescribed primarily with respect to generation of a short signature orproduct identifier, which is a code that allows validation and/orauthentication of a machine, program, user, etc. The signature is a“short” signature in that it uses a relatively small number ofcharacters.

With this in mind, attention is drawn to FIG. 1, which is a blockdiagram illustrating an exemplary cryptosystem 100 in accordance withcertain implementations of the present invention. Cryptosystem 100includes an encryptor 102 and a decryptor 104. A plaintext message 106is received at an input module 108 of encryptor 102, which is acurve-based encryptor that encrypts message 106 based on a public keygenerated based on a secret known by decryptor 104. Plaintext message106 is typically an unencrypted message, although encryptor 102 canencrypt any type of message/data. Thus, message 106 may alternatively beencrypted or encoded by some other component (not shown) or a user.

An output module 110 of encryptor 102 outputs the encrypted version ofplaintext message 106, which is ciphertext 112. Ciphertext 112 can thenbe communicated to decryptor 104, which can be implemented, for example,on a computer system remote from a computer system on which encryptor102 is implemented. Given the encrypted nature of ciphertext 112, thecommunication link between encryptor 102 and 104 need not be secure (itis typically presumed that the communication link is not secure). Thecommunication link can be any of a wide variety of public and/or privatenetworks implemented using any of a wide variety of conventional publicand/or proprietary protocols, and including both wired and wirelessimplementations. Additionally, the communication link may include othernon-computer network components, such as hand-delivery of mediaincluding ciphertext or other components of a product distributionchain.

Decryptor 104 receives ciphertext 112 at input module 114 and, beingaware of the secret used to encrypt message 106, is able to readilydecrypt ciphertext 112 to recover the original plaintext message 106,which is output by output module 116 as plaintext message 118. Decryptor104 is a curve-based decryptor that decrypts the message based on thesame curve as was used by encryptor 102.

Encryption and decryption are performed in cryptosystem 100 based on asecret, such as points on the elliptic curve. This secret is known todecryptor 104, and a public key generated based on the secret is knownto encryptor 102. This knowledge allows encryptor 102 to encrypt aplaintext message that can be decrypted only by decryptor 104. Othercomponents, including encryptor 102, which do not have knowledge of thesecret cannot decrypt the ciphertext (although decryption may betechnically possible, it is not computationally feasible). Similarly,decryptor 104 can also generate a message using the secret and based ona plaintext message, a process referred to as digitally signing theplaintext message. This signed message can then be communicated to othercomponents, such as encryptor 102, which can in turn verify the digitalsignature based on the public key.

FIG. 2 illustrates an exemplary system using a product identifier tovalidate software in accordance with certain implementations of thepresent invention. FIG. 2 illustrates a software copy generator 120including a product identifier (ID) generator 122. Software copygenerator 120 produces software media 124 (e.g., a CD-ROM, DVD (DigitalVersatile Disk)) that contains typically all the files needed tocollectively implement a complete copy of one or more applicationprograms, (e.g., a word processing program, a spreadsheet program, anoperating system, a suite of programs). These files are received fromsource files 126, which may be a local source (e.g., a hard driveinternal to generator 120), a remote source (e.g., coupled to generator120 via a network), or a combination thereof. Although only a singlegenerator 120 is illustrated in FIG. 2, typically multiple suchgenerators operate individually and/or cooperatively to increase therate at which software media 124 can be generated.

Product ID generator 122 generates a product ID 128 that can includenumbers, letters, and/or other symbols. Generator 122 generates productID 128 using the curve-based encryption techniques described herein. Theproduct ID 128 is typically printed on a label and affixed to either acarrier containing software media 124 or a box into which software media124 is placed. Alternatively, the product ID 128 may be made availableelectronically, such as a certificate provided to a user when receivinga softcopy of the application program via an on-line source (e.g.,downloading of the software via the Internet). The product ID can servemultiple functions. First, the product ID can be cryptographicallyvalidated in order to verify that the product ID is a valid product ID(and thus allowing, for example, the application program to beinstalled). Additionally, the product ID can optionally serve toauthenticate the particular software media 124 to which it isassociated.

The generated software media 124 and associated product ID 128 are thenprovided to a distribution chain 130. Distribution chain 130 representsany of a variety of conventional distribution systems and methods,including possibly one or more “middlemen” (e.g., wholesalers,suppliers, distributors, retail stores (either on-line or brick andmortar), etc.). Regardless of the manner in which media 124 and theassociated product ID 128 are distributed, eventually media 124 andproduct ID 128 are purchased (e.g., licensed), by the user of a clientcomputer 132.

Client computer 132 includes a media reader 134 capable of readingsoftware media 124 and installing the application program onto clientcomputer 132 (e.g., installing the application program on to a hard diskdrive (not shown) of client computer 132). Part of this installationprocess involves entry of the product ID 128. This entry may be a manualentry (e.g., the user typing in the product ID via a keyboard), oralternatively an automatic entry (e.g., computer 132 automaticallyaccessing a particular field of a license associated with theapplication program and extracting the product ID there from). Clientcomputer 132 also includes a product ID validator 136 which validates,during installation of the application program, the product ID 128. Thisvalidation is performed using the curve-based decryption techniques.

If validator 136 determines that the product ID is valid, then anappropriate course of action is taken (e.g., an installation program onsoftware media 124 allows the application to be installed on computer132). However, if validator 136 determines that the product ID isinvalid, then a different course of action is taken (e.g., theinstallation program terminates the installation process preventing theapplication program from being installed).

Product ID validator 136 also optionally authenticates the applicationprogram based on the product ID 128. This authentication verifies thatthe product ID 128 entered at computer 132 corresponds to the particularcopy of the application being accessed. The authentication can beperformed at different times, such as during installation, or whenrequesting product support or an upgrade. Alternatively, thisauthentication may be performed at a remote location (e.g., at a callcenter when the user of client computer 132 calls for technical support,the user may be required to provide the product ID 128 before receivingassistance).

If the application program manufacturer desires to utilize theauthentication capabilities of the product ID, then the product IDgenerated by generator 122 for each copy of an application program isunique. This uniqueness is created by assigning a different initialnumber or value to each copy of the application program. This initialvalue can then be used as a basis for generating the product ID.

The unique value associated with the copy of the application program canbe optionally maintained by the manufacturer as an authentication record138 (e.g., a database or list) along with an indication of theparticular copy of the application program. This indication can be, forexample, a serial number embedded in the application program or onsoftware media 124, and may be hidden in any of a wide variety ofconventional manners.

Alternatively, the individual number itself may be a serial number thatis associated with the particular copy, thereby allowing themanufacturer to verify the authenticity of an application program byextracting the initial value from the product ID and verifying that itis the same as the serial number embedded in the application program orsoftware media 124.

Appropriate action can be taken based on whether the product ID isauthenticated. These actions can vary, depending on the manufacturer'sdesires and/or action being taken at computer 132 that caused theauthentication check to occur. For example, if a user is attempting toinstall an application program then installation of the program may beallowed only if the authentication succeeds. By way of another example,the manufacturer's support technicians may provide assistance to a userof computer 132 only if the authentication succeeds, or an upgradeversion of the application program may be installed only ifauthentication of the previous version of the application programsucceeds.

The logic of certain curve-based cryptosystems utilizes what arecommonly referred to as “Weil and Tate pairings” during a signatureverification process when using elliptic curves. The Weil and Tatepairings have been proposed for use in many aspects of cryptography.They may be used, for example, to form efficient protocols to doone-round three-way key establishment, identity-based encryption, shortsignatures, and the like.

It is important, however, given the amount of processing to haveefficient implementations of the Weil and Tate pairings to cut down onthe cost of implementing these protocols. Computation of the Weil orTate pairing in conventional cryptosystems typically follows “Miller'salgorithm”, which is described, for example, in “Identity-BasedEncryption From The Weil Pairing”, by Dan Boneh and Matthew Franklin,published in SIAM J. of Computing, Vol. 32, No. 3, pp. 586-615, 2003.

As is well-known, for a fixed positive integer m, the Weil pairing e_(m)is a bilinear map that takes as input two m-torsion points on anelliptic curve, and outputs an m^(th) root of unity. For ellipticcurves, as is well-known, the Tate pairing is related to the Weilpairing by the fact that the Weil pairing is a quotient of the output oftwo applications of the Tate pairing. The algorithms for these pairingsdepend on constructing rational functions with a prescribed pattern ofpoles and zeros.

The Miller algorithm as typically implemented in conventionalcurve-based cryptosystems calls for the evaluation of the Weil or Tatepairing by evaluating a function at two selected points on the ellipticcurve, wherein one of the points is formed from a “random” pointselected using a randomly generated input. Unfortunately, there is achance that the Miller algorithm will essentially fail with some randominput values, due to a division by zero. When the Miller algorithmfails, then the logic will usually need to rerun the Miller algorithmusing a different random input value. Although the failure rate of theMiller algorithm tends to be fairly low, if it is required to be runthousands or millions of times, eventually the processing delays maybecome significant. Also, if the computations are performed in aparallel processing environment, the completions may be slowed ordelayed as some of the processing pipelines or the like are required torerun the Miller algorithm. Real-time applications, which have rigidbounds on maximum computation time rather than average computation time,may not be able to afford the reruns.

The improved techniques described herein provide increased efficiencyand an alternative method to the standard methods which have beenproposed. For example, in accordance with certain aspects of the presentinvention, the improved techniques do not require a randomly chosenm-torsion point as described above and as such always generate a correctanswer.

With this exemplary improvement in mind, in the following sections animproved algorithm is described for computing what is hereby referred toas the “squared Weil pairing”, e_(m)(P,Q)². With the squared Weilpairing, one may obtain a significant speed-up over the standardimplementation of the Weil pairing (Miller's algorithm). While the Tatepairing is already more efficient to implement than the Weil pairing,the improved techniques herein can also be employed in an improvedalgorithm for computing the “squared Tate pairing” for elliptic curves.The squared Weil pairing and/or the squared Tate pairing can besubstituted for the Weil or Tate pairings in any of the aboveapplications.

By way of further reference, other exemplary curve-based cryptosystemsare provided in the following references: “Short Signatures from theWeil Pairing”, by Dan Boneh, et al., in Advances inCryptography—Asiacrypt 2001, Lecture Notes in Computer Science, Vol.2248, Springer-Verlag, pp. 514-532; and, “The Weil and Tate Pairings asBuilding Blocks for Public Key Cryptosystems (Survey)”, by Antoine Joux,in Algorithmic Number Theory, 5^(th) International Symposium ANTS-V,Sydney, Australia, July 2002 proceedings, Claus Fieker and David R.Kohel (Eds.), Lecture Notes in Computer Science, Vol. 2369,Springer-Verlag, pp. 20-32.

Attention is now drawn to FIG. 3, which is a flow diagram illustratingan exemplary process 150 for use in comparing the Weil and Tate pairingsfor elliptic curves. In act 152, an addition chain, addition-subtractionchain, or the like, is formed for m, wherein m is an integer greaterthan zero and an m-torsion point P is fixed on an elliptic curve E. Inact 154, ((j+k)P, f_(j+k,P)(X)) is determined using (jP, f_(j,P)(X)) and(kP, f_(j,P)(X)), wherein j and k are integers, jP, kP and (j+k)P aremultiples of point P and f_(j,P)(X), f_(k,P)(X) and f_(j+k,P)(X) arefunctions in the indeterminate X, and ((j+k)P, f_(j+k,P)(X)) representsan iterative building block for forming the output of the pairing via achain. With Weil pairing, for example, ((j+k)P, f_(j+k,P)(X)) can alsobe run with P replaced by another m-torsion point Q, i.e., ((j+k)Q,f_(j+k,Q)(X)). In act 156, h_(j+k) is determined given h_(j) and h_(k),wherein h_(j), h_(k) and h_(j+k) are field elements and for example,

h _(j) =f _(j,P)(Q ₁)/f _(j,P)(Q ₂)  i.

for certain points Q₁ and Q₂ (independent of j) on E and the goal is tocompute h_(m). In conventional Miller algorithms Q₁ and Q₂ are randomvalue inputs.

In certain improvements provided herein, for example, an improvedalgorithm essentially produces:

h _(j) ′:=f _(jP)(Q)/f _(jP)(−Q)  a.

wherein −Q=id−Q is the complement of Q. When the elliptic curve is givenas E:y²=x³+ax+b, then the complement of a point is the reflection of thepoint over the x-axis, which for an elliptic curve is the additiveinverse of the point.

In accordance with certain aspects of the present invention, animprovement is made to act 156 wherein squared Weil pairing isintroduced. In accordance with certain further aspects of the presentinvention, an improvement is made to a subprocess of act 154 wherein aparabola is introduced for computing Weil and Tate pairings in a mannerthat reduces the number of computation steps required. Improvedcryptosystems may implement either one or both of these aspects of thepresent invention.

Squared Weil Pairing for Elliptic Curves

The purpose of this section is to construct a new pairing, referred toas squared Weil pairing, which has the advantage of being more efficientto compute than Miller's algorithm for the original Weil pairing. Theimproved algorithm presented herein has the advantage that it isguaranteed to output the correct answer since it does not depend oninputting a randomly chosen m-torsion point. As mentioned earlier,certain conventional implementations of Miller's algorithm sometimesrequire more than one iteration of the algorithm, since the randomlychosen m-torsion point may cause the algorithm to fail at times.

Let E: y²=x³+ax+b be an elliptic curve over a field K, not ofcharacteristic 2 or 3. Characteristic 2 and 3 can be handled byintroducing the more general equation for an elliptic curve.

Introducing some further notation, let:

id be the point at infinity on E;

P, Q, R, X (capitals) be points on E, wherein X is an indeterminatedenoting the (main) independent variable of a function;x(X), y(X) be (rational) functions mapping a point X on E to its(affine) x and y-coordinates;

Let line (P, Q, R)(X) denote the equation of the line (linear in x(X)and y(X)) passing through three points P, Q, R on E, which satisfyP+Q+R=id, and wherein when two of P, Q, R are equal, this is a tangentline at that common point.

Function ƒ_(j,P) and Its Construction

If j is an integer and P a point on E, then f_(j,P) and f_(j,P)(X) willrefer to a rational function on E whose divisor of zeros and poles is:

(f _(j,P))=j(P)−(jP)−(j−1)(id),

where parentheses around a point on E indicate that it is beingconsidered formally as a point on E. If j>1 and P,jP, and id aredistinct, then f_(j,P)(X) has a j-fold zero at X=P, a simple pole atX=jP, a (j−1)-fold pole at infinity (i.e., at X=id), and no other polesor zeros.

The theory of divisors states that f_(j,P) exists and is unique up to anonzero scale factor (multiplicative constant). If Q₁ and Q₂ are given,then the quotient f_(j,P)(Q₁)/f_(j,P)(Q₂) is well-defined unless adivision by zero occurs.

When j=0 or j=1, f_(j,P) can be any nonzero constant.

If one knows f_(j,P) and f_(k,P) for two integers j and k, then asimple, well-known, construction gives f_(−j−k,P). One wants f_(−j−k,P)to satisfy

(f _(−j−k,P) f _(j,P) f _(k,P))=(f _(−j−k,P))+(f _(j,P))+(f_(k,P))=3(id)−((−j−k)P)−(jP)−(kP).

This will be satisfied if we choose f_(−j−k,P) so that:

f _(−j−k,P)(X)f _(j,P)(X)f _(k,P)(X)line(jP,kP,(−j−k)P)(X)=constant.

Then repeating this construction on f_(0,P) and f_(−j−k,P) givesf_(j+k,P). The line through 0P=id, (−j−k)P, and (j+k)P is vertical(i.e., its equation does not reference the y-coordinate). This resultsin the useful constructions

${f_{{j + k},P}(X)} = {{f_{j,P}(X)}{f_{k,P}(X)}\frac{{{line}( {{jP},{kP},{( {{- j} - k} )P}} )}(X)}{{{line}( {{id},{( {{- j} - k} )P},{( {j + k} )P}} )}(X)}}$${f_{{j - k},P}(X)} = \frac{{f_{j,P}(X)}\mspace{11mu} {{line}( {{id},{jP},{- {jP}}} )}(X)}{{f_{k,P}(X)}\mspace{11mu} {{line}( {{- {jP}},{kP},{( {j - k} )P}} )}(X)}$

Other possibly useful formulae include:

f_(j,id)=constant;

f _(j,−P)(X)=f _(j,P)(−X)*(constant);

If (P+Q+R=id), then:

${{f_{j,P}(X)}{f_{j,Q}(X)}{f_{j,R}(X)}} = {\frac{{{line}( {P,Q,R} )}(X)^{j}}{{{line}( {{jP},{jQ},{jR}} )}(X)}.}$

Squared Weil-Pairing Formula

Let m be an odd prime. Suppose P and Q are m-torsion points on E, withneither being the identity and P not equal to ±Q.

Then

$\frac{{f_{m,P}(Q)}{f_{m,Q}( {- P} )}}{{f_{m,P}( {- Q} )}{f_{m,Q}(P)}} = {- {e_{m}( {P,Q} )}^{2}}$

where e_(m) denotes the Weil-pairing.

Exemplary Algorithm for e_(m)(P,Q)²

Fix an odd prime m and the curve E. Given two m-torsion points P and Qon E, one needs to compute e_(m)(P, Q)². In accordance with certainexemplary implementations of the present invention, the algorithmincludes forming an addition or addition-subtraction chain for m. Thatis, after an initial 1, every element in the chain is a sum ordifference of two earlier elements in the chain, until an m appears.Well-known techniques, give a chain of length O(log(m)). For each j inthe addition-subtraction chain, form a tuple

t_(j)=[jP,jQ,n_(j),d_(j)]

such that n_(j) and d_(j) are field elements such that

$\frac{n_{j}}{d_{j}} = {\frac{{f_{j,P}(Q)}{f_{j,Q}( {- P} )}}{{f_{j,P}( {- Q} )}{f_{j,Q}(P)}}.}$

Keeping track of the numerators and denominators separately until theend is optional.

To do this, start with t₁=[P, Q, 1, 1]. Given t_(j) and t_(k), thisprocedure gets t_(j+k):

form elliptic curve sums: jP+kP=(j+k)P and jQ+kQ=(j+k)Q;

find line: line(jP, kP, (−j−k)P)(X)=c0+c1*x(X)+c2*y(X);

find line: line(jQ, kQ, (−j−k)Q)(X)=c0′+c1′*x(X)+c2′*y(X).

Set:

n _(j+k) =n _(j) *n _(k)*(c0+c1*x(Q)+c2*y(Q))*(c0′+c1′*x(P)−c2′*y(P))and

d _(j+k) =d _(j) *d _(k)*(c0+c1*x(Q)−c2*y(Q))*(c0′+c1′*x(P)+c2′*y(P)).

A similar construction gives t_(j−k) from t_(j) and t_(k). Observe thatthe vertical lines through (j+k)P and (j+k)Q do not appear in theformulae for n_(j+k) and d_(j+k), —this is because the contributionsfrom Q and −Q (or from P and −P) are equal. Here −Q is the complement ofQ and −P is the complement of P.

When j+k=m, one can further simplify this to n_(j+k)=n_(j)*n_(k) andd_(j+k)d_(j)*d_(k), since c2 and c2′ will be zero.

Pseudocode may take the following form, for example:

i. procedure Squared_Weil_Pairing(m, P, Q) 1. issue an error if m is notan odd prime. 2. if (P = id or Q = id or P = ±Q) then a. return 1; 3.else a. t₁ = [P, Q, 1, 1]; b. use an addition-subtraction chain to geti. t_(m)=[mP, mQ, n_(m), d_(m)]. c. issue an error if mP or mQ is notid. d. if (n_(m) = 0 or d_(m) = 0) then i. return 1; e. else i. return−n_(m)/d_(m); f. end if; 4. end if;

When n_(m) and d_(m) are nonzero, then the computation

$\frac{n_{m}}{d_{m}} = \frac{{f_{m,P}(Q)}{f_{m,Q}( {- P} )}}{{f_{m,P}( {- Q} )}{f_{m,Q}(P)}}$

has been successful, and the output is correct. If, however, some n_(m)or d_(m) is zero, then some factor such as c0+c1*x(Q)+c2*y(Q) must havevanished. That line was chosen to pass through jP, kP, and (−j−k)P, forsome j and k.

This factor does not vanish at any other point on the elliptic curve.Therefore this factor can vanish only if Q=jP or Q=kP or Q=(−j−k)P forsome j and k. In all of these cases Q will be a multiple of P, ensuringthat

e _(m)(P,Q)=1.

Squared Tate Pairing For Elliptic Curves Squared Tate Pairing Formula

Let m be an odd prime. Suppose P is an m-torsion point on E, and Q is apoint on the curve, with neither being the identity and P not equal to amultiple of Q. Assume that E is defined over K, where K has q=p^(n)elements and suppose m divides q−1. Then

$( \frac{f_{m,P}(Q)}{f_{m,P}( {- Q} )} )^{\frac{q - 1}{m}} = {v_{m}( {P,Q} )}$

where v_(m) denotes the squared Tate-pairing.Exemplary Algorithm For v_(m)(P, Q)

Fix an odd prime m and the curve E. Given an m-torsion point P on E anda point Q on E, one needs to compute v_(m)(P, Q). As before, one startswith an addition or addition-subtraction chain for m. For each j in theaddition-subtraction chain, one then forms a tuple

t_(j)=[jP,n_(j),d_(j])  i.

such that

${{ii}.\mspace{14mu} \frac{n_{j}}{d_{j}}} = {\frac{f_{j,P}(Q)}{f_{j,P}( {- Q} )}.}$

Keeping track of the numerators and denominators separately until theend is optional.

Start with t₁=[P, 1, 1]. Given t_(j) and t_(k), to get t_(j+k):

-   -   i. form the elliptic curve sum jP+kP=(j+k)P;    -   ii. find line(jP, kP, (−j−k)P)(X)=c0+c1*x(X)+c2*y(X);    -   iii. set:        -   1. n_(j+k)=n_(j)*n_(k)*(c0+c1*x(Q)+c2*y(Q))            -   a. and        -   2. d_(j+k)=d_(j)*d_(k)*(c0+c1*x(Q)−c2*y(Q)).

A similar construction gives t_(j−k) from t_(j) and t_(k). Observe thatthe vertical lines through (j+k)P and (j+k)Q do not appear in theformulae for n_(j+k) and d_(j+k), because the contributions from Q and−Q are equal. When j+k=m, one can further simplify this to:

n _(j+k) =n _(i) *n _(k) and d _(j+k) =d _(j) *d _(k),

since c2 will be zero.When n_(m) and d_(m) are nonzero, then the computation

$\frac{n_{m}}{d_{m}} = \frac{f_{m,P}(Q)}{f_{m,P}( {- Q} )}$

has been successful, and after raising to the (q−1)/m power, one willhave the correct output. If, however, some n_(m) or d_(m) is zero, thensome factor such as c0+c1*x(Q)+c2*y(Q) must have vanished. That line waschosen to pass through jP, kP, and (−j−k)P, for some j and k. It doesnot vanish at any other point on the elliptic curve. Therefore thisfactor can vanish only if Q=jP or Q=kP or Q=(−j−k)P for some j and k. Inall of these cases Q will be a multiple of P.

The above techniques may be implemented through various forms of logic,including, for example, a programmed computer. Hence, FIG. 4 illustratesa more general exemplary computer environment 400, which can be used invarious implementations of the invention. The computer environment 400is only one example of a computing environment and is not intended tosuggest any limitation as to the scope of use or functionality of thecomputer and network architectures. Neither should the computerenvironment 400 be interpreted as having any dependency or requirementrelating to any one or combination of components illustrated in theexemplary computer environment 400.

Computer environment 400 includes a general-purpose computing device inthe form of a computer 402. Computer 402 can implement, for example,encryptor 102 or decryptor 104 of FIG. 1, generator 120 or clientcomputer 132 of FIG. 2, either or both of modules 152 and 153 of FIG. 3,and so forth. Computer 402 represents any of a wide variety of computingdevices, such as a personal computer, server computer, hand-held orlaptop device, multiprocessor system, microprocessor-based system,programmable consumer electronics (e.g., digital video recorders),gaming console, cellular telephone, network PC, minicomputer, mainframecomputer, distributed computing environment that include any of theabove systems or devices, and the like.

The components of computer 402 can include, but are not limited to, oneor more processors or processing units 404, a system memory 406, and asystem bus 408 that couples various system components including theprocessor 404 to the system memory 406. The system bus 408 representsone or more of any of several types of bus structures, including amemory bus or memory controller, a peripheral bus, an acceleratedgraphics port, and a processor or local bus using any of a variety ofbus architectures. By way of example, such architectures can include anIndustry Standard Architecture (ISA) bus, a Micro Channel Architecture(MCA) bus, an Enhanced ISA (EISA) bus, a Video Electronics StandardsAssociation (VESA) local bus, and a Peripheral Component Interconnects(PCI) bus also known as a Mezzanine bus.

Computer 402 typically includes a variety of computer readable media.Such media can be any available media that is accessible by computer 402and includes both volatile and non-volatile media, removable andnon-removable media.

The system memory 406 includes computer readable media in the form ofvolatile memory, such as random access memory (RAM) 410, and/ornon-volatile memory, such as read only memory (ROM) 412. A basicinput/output system (BIOS) 414, containing the basic routines that helpto transfer information between elements within computer 402, such asduring start-up, is stored in ROM 412. RAM 410 typically contains dataand/or program modules that are immediately accessible to and/orpresently operated on by the processing unit 404.

Computer 402 may also include other removable/non-removable,volatile/non-volatile computer storage media. By way of example, FIG. 4illustrates a hard disk drive 416 for reading from and writing to anon-removable, non-volatile magnetic media (not shown), a magnetic diskdrive 418 for reading from and writing to a removable, non-volatilemagnetic disk 420 (e.g., a “floppy disk”), and an optical disk drive 422for reading from and/or writing to a removable, non-volatile opticaldisk 424 such as a CD-ROM, DVD-ROM, or other optical media. The harddisk drive 416, magnetic disk drive 418, and optical disk drive 422 areeach connected to the system bus 408 by one or more data mediainterfaces 425. Alternatively, the hard disk drive 416, magnetic diskdrive 418, and optical disk drive 422 can be connected to the system bus408 by one or more interfaces (not shown).

The disk drives and their associated computer-readable media providenon-volatile storage of computer readable instructions, data structures,program modules, and other data for computer 402. Although the exampleillustrates a hard disk 416, a removable magnetic disk 420, and aremovable optical disk 424, it is to be appreciated that other types ofcomputer readable media which can store data that is accessible by acomputer, such as magnetic cassettes or other magnetic storage devices,flash memory cards, CD-ROM, digital versatile disks (DVD) or otheroptical storage, random access memories (RAM), read only memories (ROM),electrically erasable programmable read-only memory (EEPROM), and thelike, can also be utilized to implement the exemplary computing systemand environment.

Any number of program modules can be stored on the hard disk 416,magnetic disk 420, optical disk 424, ROM 412, and/or RAM 410, includingby way of example, an operating system 426, one or more applicationprograms 428, other program modules 430, and program data 432. Each ofsuch operating system 426, one or more application programs 428, otherprogram modules 430, and program data 432 (or some combination thereof)may implement all or part of the resident components that support thedistributed file system.

A user can enter commands and information into computer 402 via inputdevices such as a keyboard 434 and a pointing device 436 (e.g., a“mouse”). Other input devices 438 (not shown specifically) may include amicrophone, joystick, game pad, satellite dish, serial port, scanner,and/or the like. These and other input devices are connected to theprocessing unit 404 via input/output interfaces 440 that are coupled tothe system bus 408, but may be connected by other interface and busstructures, such as a parallel port, game port, or a universal serialbus (USB).

A monitor 442 or other type of display device can also be connected tothe system bus 408 via an interface, such as a video adapter 444. Inaddition to the monitor 442, other output peripheral devices can includecomponents such as speakers (not shown) and a printer 446 which can beconnected to computer 402 via the input/output interfaces 440.

Computer 402 can operate in a networked environment using logicalconnections to one or more remote computers, such as a remote computingdevice 448. By way of example, the remote computing device 448 can be apersonal computer, portable computer, a server, a router, a networkcomputer, a peer device or other common network node, and the like. Theremote computing device 448 is illustrated as a portable computer thatcan include many or all of the elements and features described hereinrelative to computer 402.

Logical connections between computer 402 and the remote computer 448 aredepicted as a local area network (LAN) 450 and a general wide areanetwork (WAN) 452. Such networking environments are commonplace inoffices, enterprise-wide computer networks, intranets, and the Internet.

When implemented in a LAN networking environment, the computer 402 isconnected to a local network 450 via a network interface or adapter 454.When implemented in a WAN networking environment, the computer 402typically includes a modem 456 or other means for establishingcommunications over the wide network 452. The modem 456, which can beinternal or external to computer 402, can be connected to the system bus408 via the input/output interfaces 440 or other appropriate mechanisms.It is to be appreciated that the illustrated network connections areexemplary and that other means of establishing communication link(s)between the computers 402 and 448 can be employed.

In a networked environment, such as that illustrated with computingenvironment 400, program modules depicted relative to the computer 402,or portions thereof, may be stored in a remote memory storage device. Byway of example, remote application programs 458 reside on a memorydevice of remote computer 448. For purposes of illustration, applicationprograms and other executable program components such as the operatingsystem are illustrated herein as discrete blocks, although it isrecognized that such programs and components reside at various times indifferent storage components of the computing device 402, and areexecuted by the data processor(s) of the computer.

Computer 402 typically includes at least some form of computer readablemedia. Computer readable media can be any available media that can beaccessed by computer 402. By way of example, and not limitation,computer readable media may comprise computer storage media andcommunication media. Computer storage media includes volatile andnonvolatile, removable and non-removable media implemented in any methodor technology for storage of information such as computer readableinstructions, data structures, program modules or other data. Computerstorage media include, but are not limited to, RAM, ROM, EEPROM, flashmemory or other memory technology, CD-ROM, digital versatile disks (DVD)or other optical storage, magnetic cassettes, magnetic tape, magneticdisk storage or other magnetic storage devices, or any other media whichcan be used to store the desired information and which can be accessedby computer 402. Communication media typically embody computer readableinstructions, data structures, program modules or other data in amodulated data signal such as a carrier wave or other transportmechanism and includes any information delivery media. The term“modulated data signal” means a signal that has one or more of itscharacteristics set or changed in such a manner as to encode informationin the signal. By way of example, and not limitation, communicationmedia include wired media such as wired network or direct-wiredconnection, and wireless media such as acoustic, RF, infrared and otherwireless media. Combinations of any of the above should also be includedwithin the scope of computer readable media.

The invention has been described herein in part in the general contextof computer-executable instructions, such as program modules, executedby one or more computers or other devices. Generally, program modulesinclude routines, programs, objects, components, data structures, etc.that perform particular tasks or implement particular abstract datatypes. Typically the functionality of the program modules may becombined or distributed as desired in various implementations.

For purposes of illustration, programs and other executable programcomponents such as the operating system are illustrated herein asdiscrete blocks, although it is recognized that such programs andcomponents reside at various times in different storage components ofthe computer, and are executed by the data processor(s) of the computer.

Alternatively, the invention may be implemented in hardware or acombination of hardware, software, smartcard, and/or firmware. Forexample, one or more application specific integrated circuits (ASICs)could be designed or programmed to carry out the invention.

CONCLUSION

Although the description above uses language that is specific tostructural features and/or methodological acts, it is to be understoodthat the invention defined in the appended claims is not limited to thespecific features or acts described. Rather, the specific features andacts are disclosed as exemplary forms of implementing the invention.

1. A method comprising: selecting an elliptic curve; determining aSquared Weil pairing based on the elliptic curve; and cryptographicallyprocessing selected information based on the Squared Weil pairing;wherein the elliptic curve includes an elliptic curve E over a field K,wherein E can be represented as an equation y²=x³+ax+h.
 2. The method asrecited in claim 1, wherein determining the Squared Weil pairing basedon the elliptic curve further includes establishing a point id that isdefined as a point at infinity on E, and wherein P, Q, R, X are pointson E wherein X is an indeterminate denoting an independent variable of afunction, and wherein x(X), y(X) are functions mapping the point X on Eto its affine x and y coordinates, and wherein a line passes through thepoints P, Q, R if P+Q+R=id.
 3. The method as recited in claim 2, whereinwhen at least two of the P, Q, R points are equal, the line is a tangentline at a common point.
 4. The method as recited in claim 2, whereindetermining the Squared Weil pairing based on the elliptic curve furtherincludes: with a first function ƒ_(j,P) and a second function ƒ_(k,P)for two integers j and k, deriving a third function ƒ_(−j−k,P) based onthe first and second functions.
 5. The method as recited in claim 4,wherein(f_(−j−k,P)f_(j,P)f_(k,P))=(f_(−j−k,P))+(f_(j,P))+(f_(k,P))=3(id)−((−j−k)P)−(jP)−(kP).6. The method as recited in claim 4, wherein f_(−j−k,P)(X) f_(j,P)(X)f_(k,P)(X) line(jP,kP,(−j−k)P)(X)=a constant.
 7. The method as recitedin claim 4, wherein if j is an integer and P a point on E, then thefirst and second functions are rational functions on E whose divisor ofzeros and poles is (f_(j,P))=j(P)−(jP)−(j−1)(id).
 8. The method asrecited in claim 7, wherein if j>1 and P, jP, and id are distinct, thenthe first function has a j-fold zero at X=P, a simple pole at X=jP, a(j−1)-fold pole at infinity, and no other poles or zeros.
 9. The methodas recited in claim 7, wherein if j equals 0 or 1 then the firstfunction is a nonzero constant.
 10. The method as recited in claim 4,further comprising determining f_(0,P) such that a line through 0P=id,(−j−k)P, and (j+k)P is vertical in that its equation does not referencea y-coordinate.
 11. The method as recited in claim 10, wherein:${{f_{{j + k},P}(X)} = {{f_{j,P}(X)}{f_{k,P}(X)}\frac{{{line}( {{jP},{kP},{( {{- j} - k} )P}} )}(X)}{{{line}( {{id},{( {{- j} - k} )P},{( {j + k} )P}} )}(X)}}},{and}$${f_{{j - k},P}(X)} = {\frac{{f_{j,P}(X)}{{line}( {{id},{jP},{- {jP}}} )}(X)}{{f_{k,P}(X)}{{line}( {{- {jP}},{kP},{( {j - k} )P}} )}(X)}.}$12. The method as recited in claim 10, wherein: f_(j,id)=constant;f_(j,−P)(X)=f_(j,P)(−X)*(constant); and if (P+Q+R=id), then:${{f_{j,P}(X)}{f_{j,Q}(X)}{f_{j,R}(X)}} = {\frac{{{line}( {P,Q,R} )}(X)^{j}}{{{line}( {{jP},{jQ},{jR}} )}(X)}.}$13. The method as recited in claim 2, wherein P and Q are m-torsionpoints on E and m is an odd prime, and wherein determining the SquaredWeil pairing further includes: determining the squared Weil pairingbased on${\frac{{f_{m,P}(Q)}{f_{m,Q}( {- P} )}}{{f_{m,P}( {- Q} )}{f_{m,Q}(P)}} = {- {e_{m}( {P,Q} )}^{2}}},$where e_(m) denotes the Weil-pairing.
 14. A computer-readable storagemedium having computer-implementable instructions for causing at leastone processing unit to perform acts comprising: determining a SquaredWeil pairing based on an elliptic curve; and cryptographicallyprocessing selected information based on the Squared Weil pairing;wherein the elliptic curve includes an elliptic curve E over a field K,wherein E can be represented as an equation y²=x³+ax+h.
 15. Thecomputer-readable storage medium as recited in claim 14, determining theSquared Weil pairing based on the elliptic curve further includesestablishing a point id that is defined as a point at infinity on E, andwherein P, Q, R, X are points on E wherein X is an indeterminatedenoting an independent variable of a function, and wherein x(X), y(X)are functions mapping the point X on E to its affine x and ycoordinates, and wherein a line passes through the points P, Q, R ifP+Q+R=id.
 16. The computer-readable storage medium as recited in claim15, wherein determining the Squared Weil pairing based on the ellipticcurve further includes: determining a first function ƒ_(j,P) and asecond function ƒ_(k,P) for two integers j and k; and determining athird function ƒ_(−j−k,P) based on the first and second functions. 17.The computer-readable storage medium as recited in claim 16, wherein(f_(−j−k,P)f_(j,P)f_(k,P))=(f_(−j−k,P))+(f_(j,P))+(f_(k,P))=3(id)−((−j−k)P)−(jP)−(kP).18. A method comprising: determining a Squared Weil Pairing e_(m)(P, Q)²by: establishing an odd prime m on a curve E, and based on two m-torsionpoints P and Q on E, computing e_(m)(P, Q)²; and forming a mathematicalchain for m.
 19. The method as recited in claim 18, wherein themathematical chain is selected from a group of mathematical chainscomprising an addition chain and an addition-subtraction chain; whereinin forming the mathematical chain for m, every element in themathematical chain is a sum or difference of two earlier elements in themathematical chain, which continues until m is included in themathematical chain.
 20. The method as recited in claim 18, wherein foreach j in the mathematical chain, a tuple t_(j)=[jP,jQ,n_(j),n_(d)] isformed comprising${\frac{n_{j}}{d_{j}} = \frac{{f_{j,P}(Q)}{f_{j,Q}( {- P} )}}{{f_{j,P}( {- Q} )}{f_{j,Q}(P)}}};$starting with t₁=[P, Q 1, 1], given t_(j) and t_(k), determine t_(j+k)by: forming elliptic curve sums: jP+kP=(j+k)P and jQ+kQ=(j+k)Q;determining line(jP,kP,(−j−k)P)(X)=c0+c1*x(X)+c2*y(X); determiningline(jQ,kQ,(−j−k)Q)(X)=c0′+c1′*x(X)+c2′*y(X); settingn_(j+k)=n_(j)*n_(k)*(c0+c1*x(Q)+c2*y(Q))*(c0′+c1′*x(P)−c2′*y(P)) andd_(j+k)=d_(j)*d_(k)*(c0+c1*x(Q)−c2*y(Q))*(c0′+c1′*x(P)+c2′*y(P)).